FISCHER DECOMPOSITIONS OF ENTIRE FUNCTIONS OF HILBERT–SCHMIDT HOLOMORPHY TYPE

Abstract

A Fischer pair (FP) for a vector space E is a pair (u, v) of linear maps on E, not necessarily everywhere defined, such that E= ker u⊕ Im v (Fischer decomposition). Thus, in particular, every densely defined closed operator u on a Hilbert space E forms a Fischer pair together with its adjoint u*, whenever Im u, or equivalently, Im u* is closed since then Im u*= ker u⊥. The question of when a given pair of maps (u, v) is a FP is related to the well-posedness of the (abstract) Cauchy–Goursat problem for u, v in E. We establish some Fischer pairs, for spaces that are built up by homogeneous Hilbert–Schmidt polynomials on a Hilbert space, consisting of differential and multiplication operators. In particular we study Fischer decompositions of the space of entire functions of Hilbert–Schmidt type. As a basis we generalize Fischers theorem for homogeneous polynomials in n variables to Hilbert–Schmidt polynomials.